A and B undertake to do a piece of work for rs. 600. A can do it in 10 days and B can do it in 15 days. With the help of C they finish the work in 4 days. How much C should be paid for his contribution.

Options

A. a. 400rs
B. b. 200rs
C. c. 250rs
D. d. none

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C, and then calculate how much C should be paid for his contribution.

### Step 1: Determine the work rates of A and B
- A can complete the work in 10 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ work/day} \]
- B can complete the work in 15 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{15} \text{ work/day} \]

### Step 2: Calculate the combined work rate of A and B
To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Combined work rate of A and B} = \frac{1}{10} + \frac{1}{15} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30. Therefore:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
Now, adding these gives:
\[ \text{Combined work rate of A and B} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ work/day} \]

### Step 3: Determine the combined work rate of A, B, and C
Together, A, B, and C finish the work in 4 days, so their combined work rate is:
\[ \text{Combined work rate of A, B, and C} = \frac{1}{4} \text{ work/day} \]

### Step 4: Calculate C's work rate
To find C's work rate, we subtract the combined work rate of A and B from the combined work rate of A, B, and C:
\[ \text{Work rate of C} = \frac{1}{4} - \frac{1}{6} \]
Finding a common denominator (LCM of 4 and 6 is 12):
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12} \]
Thus,
\[ \text{Work rate of C} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12} \text{ work/day} \]

### Step 5: Determine the total work done
The total work can be calculated as follows:
- A's contribution in 4 days:
\[ \text{Work done by A in 4 days} = 4 \times \frac{1}{10} = \frac{4}{10} = \frac{2}{5} \]
- B's contribution in 4 days:
\[ \text{Work done by B in 4 days} = 4 \times \frac{1}{15} = \frac{4}{15} \]
- C's contribution in 4 days:
\[ \text{Work done by C in 4 days} = 4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \]

### Step 6: Calculate the total work done by A, B, and C
Now we need to find the total work done by A, B, and C:
\[ \text{Total work} = \frac{2}{5} + \frac{4}{15} + \frac{1}{3} \]
Finding a common denominator (LCM of 5, 15, and 3 is 15):
\[ \frac{2}{5} = \frac{6}{15}, \quad \frac{1}{3} = \frac{5}{15} \]
Thus,
\[ \text{Total work} = \frac{6}{15} + \frac{4}{15} + \frac{5}{15} = \frac{15}{15} = 1 \text{ (which is the total work)} \]

### Step 7: Determine the payment for C
The total payment for the work is Rs. 600. We need to find C's share based on the work done:
- Total efficiency = A's efficiency + B's efficiency + C's efficiency:
\[ \text{Total efficiency} = 6 + 4 + 5 = 15 \]
- C's share of the payment:
\[ \text{C's share} = \frac{C's efficiency}{Total efficiency} \times Total payment = \frac{5}{15} \times 600 = \frac{1}{3} \times 600 = 200 \]

### Final Answer:
C should be paid Rs. 200 for his contribution.